Fourier Analysis and Statistics (PHYS09055)
Physics and Astronomy
Normal Year Taken
Delivery Session Year
A coherent 20pt course taken by all single honours physics students. Examined via a single three-hour paper in the December diet.
Fourier Analysis (20 lectures) - Fourier series: sin and cos as a basis set; calculating coefficients; complex basis; convergence, Gibbs phenomenon- Fourier transform: limiting process; uncertainty principle; application to Fraunhofer diffraction- Dirac delta function: Sifting property; Fourier representation- Convolution; Correlations; Parseval's theorem; power spectrum- Sampling; Nyquist theorem; data compression- Solving Ordinary Differential Equations with Fourier methods; driven damped oscillators- Green's functions for 2nd order ODEs; comparison with Fourier methods- Partial Differential Equations: wave equation; diffusion equation; Fourier solution- Partial Differential Equations: solution by separation of variables- PDEs and curvilinear coordinates; Bessel functions; Sturm-Liouville theory: complete basis set of functions Probability and Statistics (20 lectures) - Concept and origin of randomness; randomness as frequency and as degree of belief- Discrete and continuous probabilities; combining probabilities; Bayes theorem- Probability distributions and how they are characterised; moments and expectations; error analysis- Permutations, combinations, and partitions; Binomial distribution; Poisson distribution- The Normal or Gaussian distribution and its physical origin; convolution of probability distributions; Gaussian as a limiting form- Shot noise and waiting time distributions; resonance and the Lorentzian; growth and competition and power-law distributions- Hypothesis testing; idea of test statistics; chi-squared statistic; F-statistic- Parameter estimation; properties of estimators; maximum likelihood methods; weighted mean and variance; minimum chi-squared method; confidence intervals- Bayesian inference; priors and posteriors; maximum credibility method; credibility intervals- Correlation and covariance; tests of correlation; rank correlation test; least squares line fitting- Model fitting; analytic curve fitting; numerical model fitting; methods for finding minimum chi-squared or maximum credibility; multi-parameter confidence intervals; interesting and uninteresting parameters
Written Exam 80%, Coursework 20%, Practical Exam 0%
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